One-dimensional heat conduction process in the cylindrical coordinate is investigated, and a similarity type of general solution is developed using the Kummer functions. The limiting behaviors of the general solution are studied using the properties of the Kummer functions, and some useful identities are deduced. As applications of the general solution, an infinite line source problem under power-type initial and boundary conditions and a one-phase Stefan problem with space-dependent latent heat in the cylindrical coordinate are studied. The analytical solutions for these two problems are established using the general solution directly. Computational examples for the analytical solutions are presented. For the infinite line source problem, the computational results are compared with those of the solid cylindrical surface model, and the computational error caused by neglecting the radial dimension of the heat source under time-varying source intensity is investigated. For the one-phase Stefan problem, both the coefficient in the solution and the development of the temperature field are presented; the computational results can be used to verify the accuracy of numerical solutions for Stefan problems.